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Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

Mystic Rose

Age 14 to 16 Challenge Level:
How many lines are drawn at each stage of construction of the mystic rose?
Now look at the completed mystic rose - can you describe how many lines are drawn from each point?
How do these relate to "Alison's method" and "Charlie's method"?